Rotation of a weighted pulley

A light wheel of radius a has a uniform semicircular rim of mass M, and my rotate freely in a vertical plane about a horizontal axis through its center. A light string passes around the wheel and suspends a mass m. You may assume that this mechanical system is governed by the equation:
(M+m)a^2 (d^x/dt^2)= ag(m -2Msin(x)/(pi))
where x is the angle between teh downward vertical and the diameter through the center of mass of the heavy rim.

where does bifurcation occur and what type(s) of bifurcation are they? how does the wheel behave when k is large?